Topic: Higher-period ordered phases on the Bethe lattice February 21

Presenter: James Freericks, Georgetown University

Abstract: The Bethe lattice is bipartite, and hence can be easily shown to display period-two long-range-order. We illustrate how the Bethe lattice generically displays higher-period ordered phases by explicitly finding the period-three regions of stability for the spinless Falicov-Kimball model in the infinite-coordination-number limit. Surprisingly, we discover that the while the phase transition to the period-two phase is second order, the transition to a period-three phase is always first-order and occurs at very low temperatures. We also find the presence of "bound-states" in the spectrum, as the interacting density of states develops delta-function contributions for the period-three phase. Hence, even though higher-period phases do exist, they only occur at very low temperatures and in a small region of phase space.

Topic: On the smooth ergodic theory of some examples of parabolic flows February 21

Presenter: Giovanni Forni, Princeton University

Abstract: We will present recent results on the behaviour of ergodic averages of smooth functions for two examples of `parabolic' conservative flows: generic area-preserving flows (with saddle-like singularities) on higher genus surfaces and horocycle flows on (compact) surfaces of constant negative curvature. We prove that the deviation of ergodic averages from the leading behaviour determined by the ergodic theorem exhibits a power-law decay controlled by invariant distributions. In the case of flows on higher genus surfaces this result was part of a series of conjectures by M.Kontsevich and A.Zorich. The proofs are based on the analysis of the hyperbolicity properties of the appropriate `renormalization' dynamics, related to the Teichmuller flow on the moduli space in the case of flows on higher genus surfaces and to the geodesic flow in the case of horocycle flows.

Topic: Mather's theory on Lagrangian systems and connecting orbits February 22

Presenter: Jeff Xia, Northwestern University

Topic: Homotopy G-actions on spheres February 22

Presenter: Jesper Grodal, Institute for Advanced Study

Abstract: I will talk about group actions on arbitrary CW-complexes with the mod p homology of a sphere. Say that two such spaces are equivalent if they can be connected by a zig-zag of G-maps which are non-equivariant equivalences. In recent work Jeff Smith and I classify these actions in the case where G is a p-group, extending classical work on actions on finite complexes by tom Dieck, Dotzel and Hamrick. I'll discuss this result and its consequences, and maybe also discuss generalizations to arbitrary finite groups. This talk does not depend on my talks in the fall.

Topic: Group negative curvature for atoroidal 3-manifolds with essential laminations February 22

Presenter: David Gabai, Caltech

Topic: What numbers can be represented by a quadratic form? February 22

Presenter: Jonathan Hanke, Rutgers University

Topic: KKM on trees February 23

Presenter: Eli Berger, Princeton University

Abstract: Knaster-Kuratowski-Mazurkiewicz (KKM) theorem is a powerful tool in many areas of mathematics. In this talk a version of KKM theorem for trees is introduced and used to prove several combinatorial theorems. A 2-trees hypergraph is a family of nonempty subsets of the union of two trees T and R, each of which has a connected intersection with T and with R. A homogeneous 2 trees hypergraph is a family of subsets of T each of which is the union of two connected sets. For each such hypergraph H we denote by $\tau(H)$ the minimal cardinality of a set intersecting all sets in the hypergraph and by $\nu(H)$ the maximal number of disjoint sets in it. In this talk I prove that in a 2-trees hypegraph $\tau(H) \leq 2 \nu(H)$ and in a homogeneous 2-tree hypergraph $\tau(H) \

leq 3\nu(H)$. This improves the result of Alon, that $\tau(H) \leq 8 \nu(H)$ in both cases.

Topic: A characterization of isoparamteric hypersurfaces of Clifford type February 23

Presenter: Gary Jensen, Washington University in St. Louis

Topic: L^p and dispersive estimates for the wave equation with the inverse-square February 26

potential

Presenter: Shadi Tahvildar-Zadeh, Rutgers University

Topic: Stochastic Optimization Problems in Finance February 26

Presenter: Ronnie Sircar, ORFE, Princeton University

Abstract: We discuss some stochastic control problems that arise in financial applications involving derivative securities like options. These are related to state-dependent utility maximization problems in classical economics. We analyze the dual problem obtained from the Legendre transform of the associated Bellman equation and interpret the optimal strategy as the perfect hedging strategy for a modified claim. Under the assumption that volatility is random and "fast mean-reverting", we derive, using a singular perturbation analysis, approximate value functions and strategies that are easy to implement and study. The analysis identifies the usual mean historical volatility and the harmonically-averaged long-run volatility as important statistics for such optimization problems without further specification of a stochastic volatility model. We discuss some example problems such as partial hedging of derivative risk, optimal asset allocation and utility-indifference pricing, and study the effectiveness of these strategies using simulated stock paths.

Topic: Localization and Gromov-Witten theory February 26

Presenter: Rahul Pandharipande, California Institute of Technology

Abstract: I will explain the virtual localization formula for integration in Gromov-Witten theory. Localization provides a flexible tool for studying GW theory in higher genus. Three applications will be discussed: Calabi-Yau geometry, Kontsevich's combinatorial model, and the geometry of the moduli space of curves.

Topic: Ideal Membership in Polynomial Rings over the Integers: Kronecker's Problem February 27

Presenter: M. Aschenbrenner, University of Illinois at Champaign-Urbana

Topic: Exponential convergence to nonequilibrium stationary states in classical February 28

statistical mechanics

Presenter: Luc Rey-Bellet, University of Virginia

Topic: The longest paths in random matrices: connection to the eigenvalues of February 28

random matrices

Presenter: Jinho Baik, Princeton University

Abstract: In the classical central limit theorem, the sum of i.i.d. random variables has the same limiting fluctuation : the Gaussian distribution. Now we make an M by N matrix with entries of i.i.d. random variables, and take a maximum of sums of entries along a class of up/right paths. Then we ask the question of the limiting fluctuation as M and/or N tend to infinity. This type of problem arises in for example, queueing theory and interacting particle systems. There are a few examples for which one can compute the limiting distribution directly, and it turned out that the role played by the Gaussian distribution in the classical central limit theorem is now played by the largest eigenvalue of a random Hermitian matrix taken from the Gaussian unitary ensemble. We will discuss topics centered around this relation between the longest path in a random matrix and the largest eigenvalue of a random Hermitian matrix.

Topic: Noise suppression by noise March 1

Presenter: Jose Vilar, Princeton University

Abstract: Noise is usually considered to be just a source of disorder, a nuisance to be avoided. Recently, this view has been changing. There is now a growing recognition that noise in many biological and physical systems could actually make them more sensitive and efficient. I will discuss some of such situations and show how the intrinsic noise displayed by some systems can be reduced through its nonlinear interplay with externally added noise

Topic: Estimated transversality, projective maps and symplectic invariants March 1

Presenter: Denis Auroux, MIT

Abstract: The aim of this talk will be to explain how the topology of compact symplectic manifolds can be investigated using structures such as symplectic Lefschetz pencils and symplectic branched coverings over projective spaces. Focusing particularly on the four-dimensional case, we will introduce monodromy invariants and discuss some applications.

Topic: TBA March 2

Presenter: Gui Changfeng, University of Connecticut and UBC

Topic: TBA March 5

Presenter: Walter Strauss, Brown University

Topic: Absorbing Boundary Conditions for Acoustics March 5

Presenter: Jan Hesthaven, Brown University

Abstract: The numerical solution of wave-dominated problems in domains of infinite extend often require careful attention to the design and application of artificial absorbing boundary conditions to enable an accurate, efficient and robust solution of the infinite problem using a smaller finite computational domain. Although this problem is almost as old as computational modeling itself and approximate solutions numerous, it remains one of the central, yet essentially open, issues in the accurate solution of a multitude of problems in, e.g., electromagnetics, gas-dynamics, aero-acoustics, and non-linear optics. Solutions to such problems become ever more important as the development of computational methods and resources enables the high-order accurate solution of very large problems over very long periods of time where even very low levels of reflections from the artificial boundary can prohibit the expected fidelity of the solution. The 1994 introduction of the Perfectly Matched Layer (PML) methods, consisting of a sponge layer capable of absorbing all incoming waves, regardless of their frequency and angel of incidence, seemed at first to essentially eliminate this critical issue for problems of electromagnetics and, shortly thereafter, for related problems in acoustics and linear elasticity. However, subsequent analysis of these scheme has exposed many problems and many open questions to address. In this talk we shall focus the attention on the construction and analysis of PML methods for problems in acoustics. We shall begin by showing that the original approach by which the PML equations are obtained, utilizing a non-physical splitting of the equations, leads to loss of strong wellposedness of the partial differential equations and, subsequently, the possibility of exponential instability of the semi-discrete form under low-order perturbations. As we shall discuss briefly, this is a general result for splitfield formulations of PML methods as illustrated by examples from electromagnetics, acoustics, and elasticity. We continue by discussing PML schemes for the special case of ambient acoustics before addressing the more general, and much more complex, question of PML schemes for general convective aero-acoustics. Rather than using physical arguments, we present a general mathematical procedure that enables the derivation of a strongly wellposed PML scheme for the case of a constant mean flow. Computational experiments show its superior performance but also exposes a very curious problem with this, and all other PML methods, when subjected to a special excitations. We shall conclude by explaining this issue and propose a solution. This work has been done in collaboration with Saul Abarbanel (Tel Aviv University) and David Gottlieb (Brown University).

Topic: Segregation problems in binary fluids March 7

Presenter: Rafelle Esposito, University of Rome

Topic: Free probability, free entropy and applications to von Neumann algebras March 7

Presenter: Liming Ge, University of New Hampshire

Abstract: Basic results in the theory of free probability and free entropy will be explained (e.g., free central limit theorem, limit laws of random matrices, etc.). Applications of these results to the solutions of some longstanding open questions in operator algebras will be discussed.

Topic: Infinite random matrices and ergodic measures March 8

Presenter: Alexei Borodin, University of Pennsylvania

Abstract: We introduce and study a 2-parameter family of unitarily invariant probability measures on the space of infinite Hermitian matrices. We show that the decomposition of a measure from this family on ergodic components is described by a determinantal point process on the real line. The correlation kernel for this process is explicitly computed. At certain values of parameters the kernel turns into the well-known sine kernel which describes the local correlations in Circular and Gaussian Unitary Ensembles. Thus, the random point configuration of the sine process is interpreted as the random set of ``eigenvalues'' of infinite Hermitian matrices distributed according to the corresponding measure. This is a joint work with Grigori Olshanski.

Topic: TBA March 9

Presenter: Tian-Jun Li, Princeton University

Topic: Rank-two vector bundles over a general curve: A conjecture of March 12

Bertram-Feinberg-Mukai

Presenter: H. Clemens, University of Utah

Topic: TBA March 12

Presenter: Mei-Chi Shaw, University of Notre Dame

Topic: TBA March 12

Presenter: Rich Mclaughlin, University of North Carolina

Topic: Families of singular rational curves March 13

Presenter: S. Kebekus, Bayreuth

Topic: Fluctuations for stochastic lattice gases March 14

Presenter: Rosanna Marra, University of Rome

Topic: Dynamincal bounds for the Fibonacci Hamiltonian March 15

Presenter: Alexander Kiseler, University of Chicago

Abstract: We prove new lower and upper dynamical bounds for the Fibonacci operator, a popular model of one-dimensional quasicrystals. It is given by a discrete Schr\"odinger operator on $l^2(\integers),$ \[ h_v u(n) = u(n+1)+u(n-1) + \lambda ([(n+1)\omega] - [n\omega])u(n), \] where $\omega = (\sqrt{5}-1)/2$ is the golden mean. The spectrum of this operator is known to be purely singular continuous. The bounds show that dynamics is intermediate between localization and ballistic transport. Roughly speaking, the upper bound shows that a fixed part of the wavepacket at time $T$ remains inside the ball of radius $R_1(T) \sim T^{C_1 (\log \lambda)^{-1}},$ where $\lambda$ is the strength of coupling. At the same time the lower bound shows that most of the wavepacket leaves the ball of radius $R_2(T)\sim T^{C_2 (\log \lambda)^{-1}},$ $C_1>C_2$ are universal constants. The main new element of the proof is a general upper bound criterion which is derived using ideas from subordinacy theory.

Topic: Gross-Zagier formula with characters March 22

Presenter: Shou-Wu Zhang, Columbia University

Topic: TBA March 26

Presenter: Linda Rothschild, University of California-San Diego

Topic: Stochastic Growth Models on Lattices and Trees March 26

Presenter: Thomas Liggett, University of California, Los Angeles

Abstract: For the past thirty years, probabilists have studied a number of stochastic growth models that were motivated by problems in physics and biology. One of the most important of these is known as the contact process -- growth occurs as the result of "contact" with existing individuals. Such models often exhibit phase transitions, and this is the feature that leads to most of our interest in them. Until a decade ago, the contact process was studied almost exclusively on Euclidean lattices, leading to a rather complete theory in that context. Since then, it has been discovered that the behavior of the process can be quite different on exponentially growing structures such as homogeneous trees. In particular, the phase structure is richer than it is in the lattice case. In this lecture, I will briefly describe the most important results about the contact process on Z^d, and then the contrasting results for the process on a tree. I will then discuss a variant of the contact process on a tree that has the appealing property that the critical value for the phase transition can be computed explicitly. One of the ingredients in the computation is a collection of combinatorial identities satisfied by the d-ary Catalan numbers.

Topic: TBA March 27

Presenter: J. Starr, MIT

Topic: Some mathematical challenges from materials science March 28

Presenter: Jean Taylor, Rutgers University

Topic: TBA March 30

Presenter: Vincent Moncrief, Yale University

Topic: TBA April 2

Presenter: Steve Wainger, University of Wisconsin

Topic: TBA April 2

Presenter: Eric Vanden-Eijnden, CIMS, New York University

Topic: Basic facts about wavelets that the cognoscenti are sure they know April 3

but I have doubts about this

Presenter: Guido Weiss, Washington University

Topic: Holomorphic disks and invariants for 3-manifolds and smooth 4-manifolds April 4

Presenter: Zoltan Szabo, Princeton University

Abstract: We will introduce and study topological invariants for closed 3-manifolds and smooth 4-manifolds. The 3-manifold construction uses Heegaard diagrams and a version of Lagrangian Floer homology. The 4-manifold invariant uses the previous construction, a pairing on Floer-homology and a handle decomposition of the 4-manifold. We will also present some applications in three and 4-manifold topology. This is a joint result with Peter Ozsvath.

Topic: TBA April 6

Presenter: Daniel Pollack, University of Washington

Topic: Time-dependent Taylor Vortices in Wide-Gap Spherical Couette Flow April 9

Presenter: Rainer Hollerbach, Geosciences, Princeton University

Topic: TBA April 13

Presenter: Guan Bo, University of Tennessee

Topic: Creating Stability from Instability April 16

Presenter: Christopher Jones, Brown University

Abstract: The current state-of-the-art technology in optical communications is based on the use of Dispersion Managed Solitons (DMS). These propagate on fibers with dispersion compensating itself periodically. Using variational methods and averaging, a full mathematical theory for DMS will be given. Surprisingly, it is shown that the strategy can be pushed to the point where the "pulse" is oscillating between unstable states and yet remains stable itself. Another case in which two unstable objects are put together to make a stable pulse is exhibited in the FitzHugh-Nagumo system, originally derived as a model of nerve impulse propagation. While these two phenomena are unrelated, mathematically and scientifically, they both suggest that two "wrongs" can make a "right."

Topic: TBA April 23

Presenter: John Hopfield, Molecular Biology, Princeton University

Topic: TBA April 27

Presenter: Joel Hass, Institute for Advanced Study and University of California at Davis